Conservation of energy and absorption in acoustic fields for linear and nonlinear propagation

Abstract

In the present paper, the energy effects accompanying a strong sound disturbance of a medium are analyzed. The waves may be, in time, periodic — continuous or pulsed — or have the form of single pulses. The description is based on equations which are commonly applied in nonlinear acoustics. The Fourier analysis, elements of the theory of linear operators, and analytical functions are applied. A general method is given for the construction of the absorption operator in the domain of space–time coordinates (x,t), to which the small-signal absorption coefficient corresponds. By analogy to linear equations and the corresponding dispersions equations, the quasi-dispersion equations in the case of nonlinear description are introduced. Simplification of the “classical” equation of nonlinear acoustics was performed. The relations between absorption operators in the space and time domains are shown. It is demonstrated that in nonlinear interactions, where terms of such type — nonlinear function of pressure — dominate, the power (energy) of the disturbance is conserved. Just as in the linear notation, the only reason why the total power (energy) changes is linear absorption, but that one which occurs under the conditions of nonlinear propagation. In consequence, the equations of power (and energy) balancing the disturbance have the same formal shape in nonlinear and linear descriptions. The equations provide a theoretical basis for different, easier, and more accurate methods than those used previously for determination (numerical and experimental) of, e.g., the power density of heat sources generated by sound. The function of the nonlinear gain of absorption and the function of effective absorption were also introduced. On the basis of quasi-dispersion equations the phenomenon of overtone generation (not harmonics) is shortly discussed.